Vectors and the Geometry of Space

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Vector Operations

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Vector Operations

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Vector Operations

In Theorem 11.3, u is called a unit vector in the direction of v. The process of multiplying v by to get a unit vector is called normalization of v.

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The Dot Product

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The Dot Product

You have studied two operations with vectors—vector addition and multiplication by a scalar—each of which yields another vector. In this section you will study a third vector operation, called the dot product. This product yields a scalar, rather than a vector.

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The Cross Product

Projections and Vector Components

The projection of u onto v can be written as a scalar multiple of a unit vector in the direction of v .That is,

The scalar k is called the component of u in the direction of v.

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The Cross Product

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The Cross Product

A convenient way to calculate u × v is to use the following determinant form with cofactor expansion.

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The Cross Product

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The Triple Scalar Product For vectors u, v, and w in space, the dot product of u and v × w u (v × w) is called the triple scalar product, as defined in Theorem 11.9.

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The Triple Scalar Product

Planes in Space

If the vectors u, v, and w do not lie in the same plane, the triple scalar product u (v × w) can be used to determine the volume of the parallelepiped (a polyhedron, all of whose faces are parallelograms) with u, v, and w as adjacent edges, as shown in Figure 11.41. By regrouping terms, you obtain the general form of the equation of a plane in space.

Given the general form of the equation of a plane, it is easy to find a normal vector to the plane. Simply use the coefficients of x, y, and z and write n = . Figure 11.41

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Distances Between Points, Planes, and Lines

Distances Between Points, Planes, and Lines If P is any point in the plane, you can find this distance by projecting the vector onto the normal vector n. The length of this projection is the desired distance.!

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Quadric Surfaces

Cylindrical Surfaces This circle is called a generating curve for the cylinder, as indicated in the following definition.

The fourth basic type of surface in space is a quadric surface. Quadric surfaces are the three-dimensional analogs of conic sections.

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Quadric Surfaces

Quadric Surfaces

cont’d

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Quadric Surfaces

cont’d

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Surfaces of Revolution In a similar manner, you can obtain equations for surfaces of revolution for the other two axes, and the results are summarized as follows.

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Cylindrical Coordinates

Cylindrical Coordinates

The cylindrical coordinate system, is an extension of polar coordinates in the plane to three-dimensional space.

Cylindrical to rectangular:

Rectangular to cylindrical:

The point (0, 0, 0) is called the pole. Moreover, because the representation of a point in the polar coordinate system is not unique, it follows that the representation in the cylindrical coordinate system is also not unique. 92

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Spherical Coordinates

Spherical Coordinates The relationship between rectangular and spherical coordinates is illustrated in Figure 11.75. To convert from one system to the other, use the following. Spherical to rectangular:

Rectangular to spherical:

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Spherical Coordinates

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To change coordinates between the cylindrical and spherical systems, use the following.

Figure 11.75

Vector-Valued Functions

Spherical to cylindrical (r ≥ 0):

Cylindrical to spherical (r ≥ 0):

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Limits and Continuity

Space Curves and Vector-Valued Functions A

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Differentiation of Vector-Valued Functions

Differentiation of Vector-Valued Functions

The definition of the derivative of a vector-valued function parallels the definition given for real-valued functions.

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Integration of Vector-Valued Functions

Differentiation of Vector-Valued Functions

The following definition is a rational consequence of the definition of the derivative of a vector-valued function.

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Velocity and Acceleration

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Tangent Vectors and Normal Vectors

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Tangent Vectors and Normal Vectors

Tangential and Normal Components of Acceleration

In Example 2, there are infinitely many vectors that are orthogonal to the tangent vector T(t). One of these is the vector T'(t) . This follows the property T(t) T(t) = ||T(t)||2 =1

T(t) T'(t) = 0

The coefficients of T and N in the proof of Theorem 12.4 are called the tangential and normal components of acceleration and are denoted by aT = Dt [||v||] and aN = ||v|| ||T'||.

By normalizing the vector T'(t) , you obtain a special vector called the principal unit normal vector, as indicated in the following definition.

So, you can write

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Arc Length

Tangential and Normal Components of Acceleration The following theorem gives some convenient formulas for aN and aT.

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Arc Length Parameter

Figure 12.30

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Arc Length Parameter

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Functions of Several Variables

Functions of Several Variables

For the function given by z = f(x, y), x and y are called the independent variables and z is called the dependent variable.

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Limit of a Function of Two Variables

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Continuity of a Function of Two Variables

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Continuity of a Function of Three Variables

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Partial Derivatives of a Function of Two Variables

A point (x0, y0, z0) in a region R in space is an interior point of R if there exists a δ-sphere about (x0, y0, z0) that lies entirely in R. If every point in R is an interior point, then R is called open.

This definition indicates that if z = f(x, y), then to find fx you consider y constant and differentiate with respect to x. 306

Similarly, to find fy, you consider x constant and differentiate with respect to y. 312

Partial Derivatives of a Function of Two Variables

Higher-Order Partial Derivatives

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Increments and Differentials

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Differentiability This is stated explicitly in the following definition.

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Differentiability

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Approximation by Differentials

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Chain Rules for Functions of Several Variables

Chain Rules for Functions of Several Variables

The Chain Rule in Theorem 13.7 is shown schematically in Figure 13.41.

Figure 13.39

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Figure 13.41

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Directional Derivative

Implicit Partial Differentiation

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The Gradient of a Function of Two Variables

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The Gradient of a Function of Two Variables

The gradient of a function of two variables is a vector-valued function of two variables.

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Applications of the Gradient

Applications of the Gradient

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Functions of Three Variables

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Tangent Plane and Normal Line to a Surface

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Absolute Extrema and Relative Extrema

A Comparison of the Gradients ∇f(x, y) and ∇F(x, y, z)

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Absolute Extrema and Relative Extrema

Absolute Extrema and Relative Extrema

A minimum is also called an absolute minimum and a maximum is also called an absolute maximum. As in single-variable calculus, there is a distinction made between absolute extrema and relative extrema.

To locate relative extrema of f, you can investigate the points at which the gradient of f is 0 or the points at which one of the partial derivatives does not exist. Such points are called critical points of f.

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The Second Partials Test

Absolute Extrema and Relative Extrema It appears that such a point is a likely location of a relative extremum. This is confirmed by Theorem 13.16.

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The Method of Least Squares

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Lagrange Multipliers If ∇f(x, y) = λ∇g(x, y) then scalar λ is called Lagrange multiplier.

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Lagrange Multipliers

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Area of a Plane Region

Multiple Integration

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Double Integrals and Volume of a Solid Region Using the limit of a Riemann sum to define volume is a special case of using the limit to define a double integral. The general case, however, does not require that the function be positive or continuous.

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Having defined a double integral, you will see that a definite integral is occasionally referred to as a single integral. 531

Properties of Double Integrals

Double Integrals and Volume of a Solid Region A double integral can be used to find the volume of a solid region that lies between the xy-plane and the surface given by z = f(x, y).

Double integrals share many properties of single integrals.

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Evaluation of Double Integrals

Average Value of a Function For a function f in one variable, the average value of f on [a, b] is Given a function f in two variables, you can find the average value of f over the region R as shown in the following definition.

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Double Integrals in Polar Coordinates

Mass

This suggests the following theorem 14.3

A lamina is assumed to have a constant density. But now you will extend the definition of the term lamina to include thin plates of variable density. Double integrals can be used to find the mass of a lamina of variable density, where the density at (x, y) is given by the density function ρ.

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Moments and Center of Mass

14.5

By forming the Riemann sum of all such products and taking the limits as the norm of Δ approaches 0, you obtain the following definitions of moments of mass with respect to the x- and y-axes.

Surface Area

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Surface Area

Triple Integrals Taking the limit as

leads to the following definition.

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Triple Integrals

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Triple Integrals in Cylindrical Coordinates If f is a continuous function on the solid Q, you can write the triple integral of f over Q as

where the double integral over R is evaluated in polar coordinates. That is, R is a plane region that is either r-simple or θ-simple. If R is r-simple, the iterated form of the triple integral in cylindrical form is

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Triple Integrals in Spherical Coordinates

Jacobians

If (ρ, θ, φ) is a point in the interior of such a block, then the volume of the block can be approximated by ΔV ≈ ρ2 sin φ Δρ Δφ Δθ

In defining the Jacobian, it is convenient to use the following determinant notation.

Using the usual process involving an inner partition, summation, and a limit, you can develop the following version of a triple integral in spherical coordinates for a continuous function f defined on the solid region Q.

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Change of Variables for Double Integrals

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Vector Analysis

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Vector Fields

Vector Fields

Functions that assign a vector to a point in the plane or a point in space are called vector fields, and they are useful in representing various types of force fields and velocity fields.

Note that an electric force field has the same form as a gravitational field. That is,

Such a force field is called an inverse square field.

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Conservative Vector Fields

Conservative Vector Fields

Some vector fields can be represented as the gradients of differentiable functions and some cannot—those that can are called conservative vector fields.

The following important theorem gives a necessary and sufficient condition for a vector field in the plane to be conservative.

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Curl of a Vector Field

Curl of a Vector Field

The definition of the curl of a vector field in space is given below.

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Divergence of a Vector Field

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Divergence of a Vector Field

You have seen that the curl of a vector field F is itself a vector field. Another important function defined on a vector field is divergence, which is a scalar function.

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Objectives

15.2

Understand and use the concept of a piecewise smooth curve.

Line Integrals

Write and evaluate a line integral. Write and evaluate a line integral of a vector field. Write and evaluate a line integral in differential form.

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Line Integrals

Line Integrals

Note that if f(x, y, z) = 1, the line integral gives the arc length of the curve C. That is, 709

Line Integrals of Vector Fields

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Fundamental Theorem of Line Integrals

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Independence of Path

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Independence of Path A curve C given by r(t) for a ≤ t ≤ b is closed if r(a) = r(b). By the Fundamental Theorem of Line Integrals, you can conclude that if F is continuous and conservative on an open region R, then the line integral over every closed curve C is 0.

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Green’s Theorem

Green’s Theorem Among the many choices for M and N satisfying the stated condition, the choice of M = –y/2 and N = x/2 produces the following line integral for the area of region R.

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Normal Vectors and Tangent Planes

Parametric Surfaces

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Area of a Parametric Surface

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Surface Integrals

The area of the parallelogram in the tangent plane is ||∆uiru × ∆virv|| = ||ru × rv|| ∆ui∆vi which leads to the following definition.

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Divergence Theorem

Flux Integrals

With these restrictions on S and Q, the Divergence Theorem is as follows.

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Stokes’s Theorem

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